GATE CS 2025 Set 2 Q34 — Linear Algebra: Matrices, Determinants, System of Linear Equations

We analyze each statement based on the properties of the LU decomposition $P = LU$:

1. Statement A is TRUE: Substituting $P = LU$ into the system $PX = Q$ gives $L(UX) = Q$. By setting $UX = Y$, we can solve the system in two steps: first solve $LY = Q$ for $Y$ (via forward substitution), and then solve $UX = Y$ for $X$ (via back substitution).
2. Statement B is TRUE: Since $L$ is a unit lower triangular matrix, its determinant is $\det(L) = 1$. Using the determinant product rule, $\det(P) = \det(L)\det(U) = \det(U)$. If $P$ is invertible, then $\det(P) \neq 0$, which implies $\det(U) \neq 0$. Thus, both $L$ and $U$ are invertible.
3. Statement C is TRUE: If $P$ is singular, then $\det(P) = 0$. Since $\det(P) = \det(U)$, we have $\det(U) = 0$. For an upper triangular matrix $U$, the determinant is the product of its diagonal elements: $\det(U) = u_{11}u_{22}u_{33} = 0$. This implies at least one diagonal element of $U$ must be zero.
4. Statement D is FALSE: If $P$ is symmetric, $L$ and $U$ are generally not symmetric. Since $L$ is strictly lower triangular (with $1$s on the diagonal) and $U$ is upper triangular, they can only be symmetric if they are diagonal matrices, which is not true for general symmetric matrices. Instead, for a symmetric matrix, we have $U = DL^T$ where $D$ is a diagonal matrix.